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The method of coordinate averaging
The method of coordinate averaging Richard F. Melka (*)
ABSTRACT In this paper a p e r t u r b a t i o n t e c h n i q u e for nearly linear oscillatory systems is developed and a set o f second order averaged equations is obtained. The standard form treated is dx _ F (x, t; e), leJ ~ 1, and a well k n o w n example is considered in detail so as to show h o w dt the a s y m p t o t i c a p p r o x i m a t i o n s o f o t h e r m e t h o d s can be obtained.
1. INTRODUCTION
d_~ = eG(y, t; e) dt
Many problems in non-linear oscillations are phrased to involve a small parameter e so that advantage may be taken of asymptotic methods. Poincar~ [1] used the idea of perturbations to obtain convergent asymptotic approximations of periodic solutions. The asymptotic methods of Bogoliubov and Mitropolsky [2], Cole and Kervokian [3], Struble [4], and Morrison [5] were developed using the perturbation method. There have been slight differences in the approaches of these researchers, and their techniques generate slightly different approximations in the class of problems concerning non-linear oscillations. We show that these approximations are asymptotically equivalent by investigating a standard example. The various techniques, including the one to be developed, amount to using coordinate perturbations and variation of parameters, either implicitly or explicitly. A certain arbitrariness is present in the development of the theory of averaging. This has been used to compare averaging with the two variable expansion procedure [5] and to preserve canonical form [6]. Our technique shows that this arbitrariness can be associated with a linearization of the averaged equations. The following system of equations (standard form) is considered : dx dt F ( x , t ; e ) , l e l ~ l , x--(x I .... Xn) F (x, t + 2¢r; e) = F (x, t; e),
F = (F1 .... Fn)
(1.1) We assume that x (t; e) are a-priori bounded, while the functions F (x, t; e) are analytic in x and e, and continuous in t. Equations (1.1) are dealt with directly just as in the method developed by Poincar~. They can, however, be transformed into
(1.2)
by using the generating solutions (those corresponding to e = 0) of (1.1) as a change of variable. This is usually done in order to obtain a form to which averaging can be applied. Our technique, however, can be applied to the standard form so that approximations will be in the original coordinates; i.e. the technique yields a set of averaged equations and coordinate perturbations. Since the standard form is considered more general than (1.2), there will be more algebraic computations in the initial development; however, considerable simplification occurs and general expressions that include necessary second order effects in the parameter e ensue. 2, APPROXIMATION TO ORDER e We now consider equations (1.1) in the expanded form dX -=F0 (x, t) + eF1 (x, t) + e2F2 (x, t) + ...... (2.1) dt The generating solution is obtained by setting e = 0. The functions x (t, e = 0) are denoted by ~, so that d~ = F 0 (~, t) dt Integration of (2.2) yields
(2.2)
= ~ (a, t) = ~ (a, t + 27r) where a = (a I .... an)
(2.3)
are constants of integration. The 0 (e) contributions will be obtained by considering the expansion x = ~ (a, t) + e ¢/1 (a, t) (2.4) where the a k are taken to be slowly varying parameters. Substituting (2.4) into (2.1), and retaining terms linear in e, yields
(*) Richard F. Melka, Assistant Professor, University o f Pittsburgh at Bradford Journal of Computational and Applied Mathematics, volume 4, no 1, 1978.
41
a} d a _ e a F 0 ( } , t ) 7 l ( a , t ) _ e a 7 1 ( a ' t ) + F l(},t) aa dt a~ at
Integrating (2.12) the a k are kept constant just as in the averaging process. Hence, we have
(2.5) where
aa }(aa(a,r) F 1 [~(a,r),r]-C(a)} dr + D(a) a~ 771 =-''a~
a~aF0 ( ~ , t )
signifies axaF0(x,t) I x=
(2.13) or
We now solve equation (2.5) for - ~ - .
Noting that
a__~_~and a_aa are inverse matrices so that aa a~
a~ aa
a a _ aa a~_ I a~ a~ aa
(2.6)
the identity matrix. Hence, da
{ a a [ a F . _0¢/1
dT =e - ~ a }
aa aT/1+ aa F 1 a} at ~}
]
a
aa
a
(2.14)
If the functions D (a) were chosen indiscriminately, say as integration constants, secular effects might be introduced in the second order approximation. We therefore consider the 0 (e 2) approximation in order to determine what effect D(a) has upon the equation given in (2.11).
(2.7) 3. THE SECOND ORDER APPROXIMATION
From (2.6) we have a~
7/1 = - ~ a~ aa F 1 -C(a)} dr + aa a~ D(a) - a } {a~
We introduce the change of variable
lamina
x -- ~ (a, t) + e71 (a, t) + e27 2 (a, t)
a a~ aa[W ]
aa_
aF
a~
aa ~
aa_
aF 0
3}
Therefore a__ aa _
aa BF0
(2.8)
Equations (2.7) may now be summarized in the form da = - -aT a [[-~-~ aa 71] + ea-aaa -~~ F1
a a F 1 = C (a) + terms periodic in t.
a~
where 771 is determined by equations (2.14). We note that we are not interested in explicitly determining 7/2 but only in incorporating those terms into (2.11) that would cause unboundedness in 7 2. Substituting (3.1) into (2.1) and retaining terms to 0 (e 2) we have da dt
-
C ( a ) - e 2 a {aa ~-t ~
(2.10)
Setting 71 = 0 in (2.9) is equivalent to performing a variation of parameters technique on equation (2.1). Generally C(a) does not vanish and gives rise to secular terms. The method developed by Poincar6 enables periodic solutions to be obtained, and this corresponds to that approximation where specific values of a are chosen so as to constrain C(a) to vanish. In the two variable method [3], and the general asymptotic method in [4], the C(a) appear as non-zero coefficients of the resonant causing terms. Mt (T-~aaF 1} = ~ 1 }Tr a~aa(a'r)Fl[~(a'r)'r]dz=C(a) The a k are held constant during averaging, and we note that the secular effect that is generated in integrating (2.10) will not affect (2.9) when the latter is separated into da at - eC(a) (2.11)
+ e 2 aa
a~
{a_8~71} + ,faal~l - ~ _ -C
(a) } = 0
_
(2.12)
a2F 0 {a g
~ a~p
~
_
_
_
aF 1
1+ g 1 + F 2} riP a~p 7p K (3.2a)
Repeated indices indicate that the usual summation convention is being used. This will ensure correct interpretation of our formulas. The effects of the terms involving D (a) are to be explicitly exhibited. Hence, we rewrite 771 71 = ~1+ ~---~---Dg a aK
where the ~1 are defined by (2.13) when D = 0. Substituting this expression into (3.2), we obtain the form that contains the 0(e 2) corrections to (2.11) . d a _ c ( a ) _ e 2 a aa _21 dt ~ t ( a-~-g 'tK' e 2 aa 32F0 3~g 3~v + 2 a~ct B~KB~V an/3 nap D/aDP 1 aa + e 2{~ a~a aa
a at
72 aa at/1 g } - e 2 a~K aa a Ca
(2.9)
The last term in (2.9) can be expanded in a Fourier Series
(3.1)
(3.2b)
a 2FO
a~Ka~v[~I ~a~v+ n v ~1 a~g ]
aF I a~g
+B~T B~K aa/~
Journal of Computational and Applied Mathematics, volume 4, no 1, 1978.
aa
a2~o.
a~a aap~)a~Cp}D~ + 42
+
-a~a
a~g rig 2
a 2 Fa0
= a with o_g_= I the identity matrix. This situation aa corresponds to solving equation (1.2).
~1 ~1
a~Ka~V~K %
~1
_at/a /
CK}_e2 aD (3.2b)
The terms multiplying D/~ in (3.2b) can be simplified by applying identities A5, A3, A4, and A6, and this results in
4. Example We consider the van der Pol equation. It has been treated by various authors and their methods yield slightly different perturbation approximations.
Xl = x2
a {aa a2~a aap ~1 at a~a aa(3aap a~v ~v}
aC
3a[3
This form is preferred because all the terms in the bracket have period 2n and there will be no contribution from averaging the partial derivative of those terms with respect to t. The term multiplying D3Dp is changed for a similar reason and is noted in A4. We can now rewrite (3.2b) into a form suitable for averaging. d ~ - = C ( a ) - e 2 ~ at {~K - + e2 a
aa
0 < e 41
/¢2 = -Xl'+ e(l -x~) x 2
2
(4.1)
Applying our technique results in e 3 x I = a I cos (t-a2) - ~ a 1 sin 3(t-a2) +eD 1cos(t-a2) +eal (D2 +
(al2 -4) 16
) sin (t-a2)
3 3 x 2 = -a 1 sin (t-a2) - - ~ e a 1 cos 3 (t- a2) -eD 1 sin (t -a2) (a12 - 4) + e a 1 {D2 - } cos(t-a2). 16
nK }
a2~a
(4.2)
-2 0-7 {a~--a aa~Oap } D/~Dp
The averaged equations are
e2 { aC %
2 e2 al 2 aD1 + -(4 - g -3al)2 ~ D1} a 1 = e a l ( 4 - a 1) + {--if- (4-al) aa 1 (4.3) I
+
a aa at [-a-~a
_e2 aa aD g cK+e2 1
a2F0
a2~a aa/3aa p
1 {F2+ aFa_~l 7tK
a~;~-a aa
aap ~1]} D/~ a~v
e 2 e 2 al ~2 =~-~ + { 8
~
~1 ar/~a 1
+ 2 a~K a~v
(3.2c)
We def'me by H(a), the average value of the last four terms in (3.2c) and note that the functions 72 will remain bounded if we separate (3.2c) as indicated in section 2. Hence, equation (2.11) is now corrected to 0(e 2) : da d-~ = eC(a) + e 2 {~aC Dp + H (a) - aaD ap Cp} (3.3)
11 2, aD2+ 1 (2___~__al)}(4_a~) aa I 128 (4.4)
Equations (4.2) indicate that acceptable choices for D(a) are functions that remain bounded in t, and concurrently allow the integration of (4.3) and (4.4). Hence, we remove the 0 (e 2) terms in (4.3) and this results in D I = D 0 a 1 ( 4 - a 2) with
(4.5)
dal al 2 dt - e - ~ ( 4 - a l )
(4.6)
2 al =
4 _ 1+[4/(a )2 1]e - e t 0
(4.7)
If each D were chosen zero, equations (3.3) would read
Equation (4.4) can be integrated using (4.6), and
d__b_b= e C (b) + 62 H (b) with ~ = ~ (b, t). dt
e2 0 1 11 2~ 0 a2 = ]-~ t + e {-D 2 + D 2 + ~-ln ]all-~-~al) + a 2
(3.4)
Equations (3.3) are a linearization of (3.4) and this can be verified by letting b = a + eD(a)
(3.5)
and substituting (3.5) into (3.4) while retaining only terms to 0 (e2). In the case of F 0 (x, t) = 0, the e = 0 solutions of the system correspond to constant solutions
(4.8) The various methods assume Di to be dependent only upon the amplitude al, and D0, a 0i are constants of integration. The Cole, Kervorkian solution [3] consists of (4.2), (4.5), (4.7), and
Journal of Computational and Applied Mathematics, volume 4, no 1, 1978.
43
11 2 0 D2=~1 In {a 1 i--~al+D 2
e2
a2=~t+
tion (2.10) to be
0
a2
~}_&aF 1 = 2 ; _ Cm t > 0 or a a F 1 = 2 ; Q m e - m t at tm ' , at
(4.9)
The terms that would yield secular effects can be correspondingly put into (2.11) by choices
Struble's method [4] does not restrict the frequency a 2 to consist only of secular terms and the homogeneous solution is not added at each perturbation step. Equation (4.2) would indicate the choices : (al2 - 4) D1 = 0 D216 (4.10)
da-e{C0(a)+ dt
1 In a I 7 2 0 e{g I I --5-4 a l } + a2
Then t l = a I cos t + a 2 sin t. A straightforward perturbation yields 71 = 0 (In t) as t -* *0 ; however, the indicated procedure yields
(4.11)
We have shown in section 3 that D 1 = D 2 = 0 involves no loss of generality; hence our perturbation technique yields another valid approximation and it consists of (4.2), (4.7), and 62 1 In 11 a2= 1--6t+e(~tal[-~a
2~ 0 1J+a 2
or d-~a=eQ0(a ) dt
If t > 0, an example for the first case is given by x2 5¢1= x2 5¢2=-x1-e t
Hence, the Struble solution consists of (4.2), (4.7), (4.10), and e2 a2=-~t+
Cl(a) } t
d a _ _ e _ _a 2 dt 2t
da _ _ e a l dt 2t
and the improved approximation is t l = a 0 { t } - e / 2 c o s t + a 02 {t} - e / 2 s .m t .
(4.12)
We finally note that the results of sections 2 and 3 can be extended to higher orders and the short periodic effects are used as a transformation set to yield corrections to the averaged equations. The justification of the perturbation method is given in [2] and therefore is not repeated here.
Each of the three approximations reduces to the same periodic solution when t -* ~ or when 0 a 2 = a 2 = 2. Table 1 indicates how closely in numerical agreement these various solutions are. The numerical integration is over the time interval [0, 1] with e = . 2. The quantities a x I and Ax 2 indicate the differences between numerical integration and the indicated approximation.
The author expresses his appreciation to prof. E. Cumberbatch of Purdue University for his help in preparing this paper. APPENDIX 1 - IDENTITIES
5. CONCLUSIONS
a In the derivations of section 2 and 3, the periodic nature of (1.1) enabled the averaged equations to be reduced to a simplified form. The spirit of the method, however, can be used independently of averaging on systems known to be a priori bounded. Consider equa-
~- aa }
aa
3F~
(A1) (A2)
C ( a ) : M t ()~p-~pF1)
Table 1. Comparison with numerical integration Xl(0)
Xl(1)
Numerical integration
.99747
.51735
Cole Kervorkian
.99747
Strubh D1 = D2 = 0
ax 1
x2(0)
x2(1)
0
-.15541
-.88529
0
.51503
.00232
-.15541
-.88118
-.00411
.99747
.51675
.0006
-.15541
-.88548
-.00019
.99747
.51565
.0017
-.15541
-.88246
-.0028-3
Journal of Computational and Applied Mathematics, volume 4, no 1, 1978.
ax 2
44
C(a) -- - aa ~ - ( ~a a
a
a___a_aa2~p ) -
771} + a---~ aa Fp1
aa
a~v a~pa~/3
a.__ { a_~.a} = _ aa
a2~v
aap
aa
aaaap
a~
a~v
a~p
a~fl
aa
aa
(A4)
(A5)
a2~0 a~-----v a~pa~ aa = a~v
a 2F0 v a~pa~/3
aa
+
a~/3 ~1 aa
~P
REFERENCES
1. POINCARI~, H. : "Sur les courbes par les 6quations diff& rentielles", J. de Math, (3) 7 (1886).
a 2F0 v
at ( a ~ p aaaa
a~
(A3)
aa (A6)
2. BOGOLIUBOV, MITROPOLSKY : "Asymptotic methods in the theory of non-linear oscillations", Gordon and Breach, New York, 1961. 3. COLE, KERVORKIAN : "Uniformly valid approximations for certain non-linear differential equations". Proceedings of the 1961 International Symposium on Non-Linear Differential Equations in Non-linear Mechanics, LaSalle and Lefschetz editors, Academic Press, New York, 1963, pp. 113-120. 4. STRUBLE, R. A. : "Non-linear differential equations", Mc Graw Hill, Inc. 1962. 5. MORRISON, J. A. : "Comparison of the modified method of averaging and the two variable expansion procedure", SIAM Review, Vol. 8, No. 1, 1966. 6. BURHSTEIN, SOLOV'EV : "Hamiltonian of averaged
motion", Soviet Physics Dokl, 6 (1962), pp. 731-733. 7. MELKA, R. F. : "Perturbation methods in ordinary differential equations", Ph.D. Thesis, Purdue Univ., August 1969.
Journal o f Computational and Applied Mathematics, volume 4, no 1, 1978.
45
ABSTRACT In this paper a p e r t u r b a t i o n t e c h n i q u e for nearly linear oscillatory systems is developed and a set o f second order averaged equations is obtained. The standard form treated is dx _ F (x, t; e), leJ ~ 1, and a well k n o w n example is considered in detail so as to show h o w dt the a s y m p t o t i c a p p r o x i m a t i o n s o f o t h e r m e t h o d s can be obtained.
1. INTRODUCTION
d_~ = eG(y, t; e) dt
Many problems in non-linear oscillations are phrased to involve a small parameter e so that advantage may be taken of asymptotic methods. Poincar~ [1] used the idea of perturbations to obtain convergent asymptotic approximations of periodic solutions. The asymptotic methods of Bogoliubov and Mitropolsky [2], Cole and Kervokian [3], Struble [4], and Morrison [5] were developed using the perturbation method. There have been slight differences in the approaches of these researchers, and their techniques generate slightly different approximations in the class of problems concerning non-linear oscillations. We show that these approximations are asymptotically equivalent by investigating a standard example. The various techniques, including the one to be developed, amount to using coordinate perturbations and variation of parameters, either implicitly or explicitly. A certain arbitrariness is present in the development of the theory of averaging. This has been used to compare averaging with the two variable expansion procedure [5] and to preserve canonical form [6]. Our technique shows that this arbitrariness can be associated with a linearization of the averaged equations. The following system of equations (standard form) is considered : dx dt F ( x , t ; e ) , l e l ~ l , x--(x I .... Xn) F (x, t + 2¢r; e) = F (x, t; e),
F = (F1 .... Fn)
(1.1) We assume that x (t; e) are a-priori bounded, while the functions F (x, t; e) are analytic in x and e, and continuous in t. Equations (1.1) are dealt with directly just as in the method developed by Poincar~. They can, however, be transformed into
(1.2)
by using the generating solutions (those corresponding to e = 0) of (1.1) as a change of variable. This is usually done in order to obtain a form to which averaging can be applied. Our technique, however, can be applied to the standard form so that approximations will be in the original coordinates; i.e. the technique yields a set of averaged equations and coordinate perturbations. Since the standard form is considered more general than (1.2), there will be more algebraic computations in the initial development; however, considerable simplification occurs and general expressions that include necessary second order effects in the parameter e ensue. 2, APPROXIMATION TO ORDER e We now consider equations (1.1) in the expanded form dX -=F0 (x, t) + eF1 (x, t) + e2F2 (x, t) + ...... (2.1) dt The generating solution is obtained by setting e = 0. The functions x (t, e = 0) are denoted by ~, so that d~ = F 0 (~, t) dt Integration of (2.2) yields
(2.2)
= ~ (a, t) = ~ (a, t + 27r) where a = (a I .... an)
(2.3)
are constants of integration. The 0 (e) contributions will be obtained by considering the expansion x = ~ (a, t) + e ¢/1 (a, t) (2.4) where the a k are taken to be slowly varying parameters. Substituting (2.4) into (2.1), and retaining terms linear in e, yields
(*) Richard F. Melka, Assistant Professor, University o f Pittsburgh at Bradford Journal of Computational and Applied Mathematics, volume 4, no 1, 1978.
41
a} d a _ e a F 0 ( } , t ) 7 l ( a , t ) _ e a 7 1 ( a ' t ) + F l(},t) aa dt a~ at
Integrating (2.12) the a k are kept constant just as in the averaging process. Hence, we have
(2.5) where
aa }(aa(a,r) F 1 [~(a,r),r]-C(a)} dr + D(a) a~ 771 =-''a~
a~aF0 ( ~ , t )
signifies axaF0(x,t) I x=
(2.13) or
We now solve equation (2.5) for - ~ - .
Noting that
a__~_~and a_aa are inverse matrices so that aa a~
a~ aa
a a _ aa a~_ I a~ a~ aa
(2.6)
the identity matrix. Hence, da
{ a a [ a F . _0¢/1
dT =e - ~ a }
aa aT/1+ aa F 1 a} at ~}
]
a
aa
a
(2.14)
If the functions D (a) were chosen indiscriminately, say as integration constants, secular effects might be introduced in the second order approximation. We therefore consider the 0 (e 2) approximation in order to determine what effect D(a) has upon the equation given in (2.11).
(2.7) 3. THE SECOND ORDER APPROXIMATION
From (2.6) we have a~
7/1 = - ~ a~ aa F 1 -C(a)} dr + aa a~ D(a) - a } {a~
We introduce the change of variable
lamina
x -- ~ (a, t) + e71 (a, t) + e27 2 (a, t)
a a~ aa[W ]
aa_
aF
a~
aa ~
aa_
aF 0
3}
Therefore a__ aa _
aa BF0
(2.8)
Equations (2.7) may now be summarized in the form da = - -aT a [[-~-~ aa 71] + ea-aaa -~~ F1
a a F 1 = C (a) + terms periodic in t.
a~
where 771 is determined by equations (2.14). We note that we are not interested in explicitly determining 7/2 but only in incorporating those terms into (2.11) that would cause unboundedness in 7 2. Substituting (3.1) into (2.1) and retaining terms to 0 (e 2) we have da dt
-
C ( a ) - e 2 a {aa ~-t ~
(2.10)
Setting 71 = 0 in (2.9) is equivalent to performing a variation of parameters technique on equation (2.1). Generally C(a) does not vanish and gives rise to secular terms. The method developed by Poincar6 enables periodic solutions to be obtained, and this corresponds to that approximation where specific values of a are chosen so as to constrain C(a) to vanish. In the two variable method [3], and the general asymptotic method in [4], the C(a) appear as non-zero coefficients of the resonant causing terms. Mt (T-~aaF 1} = ~ 1 }Tr a~aa(a'r)Fl[~(a'r)'r]dz=C(a) The a k are held constant during averaging, and we note that the secular effect that is generated in integrating (2.10) will not affect (2.9) when the latter is separated into da at - eC(a) (2.11)
+ e 2 aa
a~
{a_8~71} + ,faal~l - ~ _ -C
(a) } = 0
_
(2.12)
a2F 0 {a g
~ a~p
~
_
_
_
aF 1
1+ g 1 + F 2} riP a~p 7p K (3.2a)
Repeated indices indicate that the usual summation convention is being used. This will ensure correct interpretation of our formulas. The effects of the terms involving D (a) are to be explicitly exhibited. Hence, we rewrite 771 71 = ~1+ ~---~---Dg a aK
where the ~1 are defined by (2.13) when D = 0. Substituting this expression into (3.2), we obtain the form that contains the 0(e 2) corrections to (2.11) . d a _ c ( a ) _ e 2 a aa _21 dt ~ t ( a-~-g 'tK' e 2 aa 32F0 3~g 3~v + 2 a~ct B~KB~V an/3 nap D/aDP 1 aa + e 2{~ a~a aa
a at
72 aa at/1 g } - e 2 a~K aa a Ca
(2.9)
The last term in (2.9) can be expanded in a Fourier Series
(3.1)
(3.2b)
a 2FO
a~Ka~v[~I ~a~v+ n v ~1 a~g ]
aF I a~g
+B~T B~K aa/~
Journal of Computational and Applied Mathematics, volume 4, no 1, 1978.
aa
a2~o.
a~a aap~)a~Cp}D~ + 42
+
-a~a
a~g rig 2
a 2 Fa0
= a with o_g_= I the identity matrix. This situation aa corresponds to solving equation (1.2).
~1 ~1
a~Ka~V~K %
~1
_at/a /
CK}_e2 aD (3.2b)
The terms multiplying D/~ in (3.2b) can be simplified by applying identities A5, A3, A4, and A6, and this results in
4. Example We consider the van der Pol equation. It has been treated by various authors and their methods yield slightly different perturbation approximations.
Xl = x2
a {aa a2~a aap ~1 at a~a aa(3aap a~v ~v}
aC
3a[3
This form is preferred because all the terms in the bracket have period 2n and there will be no contribution from averaging the partial derivative of those terms with respect to t. The term multiplying D3Dp is changed for a similar reason and is noted in A4. We can now rewrite (3.2b) into a form suitable for averaging. d ~ - = C ( a ) - e 2 ~ at {~K - + e2 a
aa
0 < e 41
/¢2 = -Xl'+ e(l -x~) x 2
2
(4.1)
Applying our technique results in e 3 x I = a I cos (t-a2) - ~ a 1 sin 3(t-a2) +eD 1cos(t-a2) +eal (D2 +
(al2 -4) 16
) sin (t-a2)
3 3 x 2 = -a 1 sin (t-a2) - - ~ e a 1 cos 3 (t- a2) -eD 1 sin (t -a2) (a12 - 4) + e a 1 {D2 - } cos(t-a2). 16
nK }
a2~a
(4.2)
-2 0-7 {a~--a aa~Oap } D/~Dp
The averaged equations are
e2 { aC %
2 e2 al 2 aD1 + -(4 - g -3al)2 ~ D1} a 1 = e a l ( 4 - a 1) + {--if- (4-al) aa 1 (4.3) I
+
a aa at [-a-~a
_e2 aa aD g cK+e2 1
a2F0
a2~a aa/3aa p
1 {F2+ aFa_~l 7tK
a~;~-a aa
aap ~1]} D/~ a~v
e 2 e 2 al ~2 =~-~ + { 8
~
~1 ar/~a 1
+ 2 a~K a~v
(3.2c)
We def'me by H(a), the average value of the last four terms in (3.2c) and note that the functions 72 will remain bounded if we separate (3.2c) as indicated in section 2. Hence, equation (2.11) is now corrected to 0(e 2) : da d-~ = eC(a) + e 2 {~aC Dp + H (a) - aaD ap Cp} (3.3)
11 2, aD2+ 1 (2___~__al)}(4_a~) aa I 128 (4.4)
Equations (4.2) indicate that acceptable choices for D(a) are functions that remain bounded in t, and concurrently allow the integration of (4.3) and (4.4). Hence, we remove the 0 (e 2) terms in (4.3) and this results in D I = D 0 a 1 ( 4 - a 2) with
(4.5)
dal al 2 dt - e - ~ ( 4 - a l )
(4.6)
2 al =
4 _ 1+[4/(a )2 1]e - e t 0
(4.7)
If each D were chosen zero, equations (3.3) would read
Equation (4.4) can be integrated using (4.6), and
d__b_b= e C (b) + 62 H (b) with ~ = ~ (b, t). dt
e2 0 1 11 2~ 0 a2 = ]-~ t + e {-D 2 + D 2 + ~-ln ]all-~-~al) + a 2
(3.4)
Equations (3.3) are a linearization of (3.4) and this can be verified by letting b = a + eD(a)
(3.5)
and substituting (3.5) into (3.4) while retaining only terms to 0 (e2). In the case of F 0 (x, t) = 0, the e = 0 solutions of the system correspond to constant solutions
(4.8) The various methods assume Di to be dependent only upon the amplitude al, and D0, a 0i are constants of integration. The Cole, Kervorkian solution [3] consists of (4.2), (4.5), (4.7), and
Journal of Computational and Applied Mathematics, volume 4, no 1, 1978.
43
11 2 0 D2=~1 In {a 1 i--~al+D 2
e2
a2=~t+
tion (2.10) to be
0
a2
~}_&aF 1 = 2 ; _ Cm t > 0 or a a F 1 = 2 ; Q m e - m t at tm ' , at
(4.9)
The terms that would yield secular effects can be correspondingly put into (2.11) by choices
Struble's method [4] does not restrict the frequency a 2 to consist only of secular terms and the homogeneous solution is not added at each perturbation step. Equation (4.2) would indicate the choices : (al2 - 4) D1 = 0 D216 (4.10)
da-e{C0(a)+ dt
1 In a I 7 2 0 e{g I I --5-4 a l } + a2
Then t l = a I cos t + a 2 sin t. A straightforward perturbation yields 71 = 0 (In t) as t -* *0 ; however, the indicated procedure yields
(4.11)
We have shown in section 3 that D 1 = D 2 = 0 involves no loss of generality; hence our perturbation technique yields another valid approximation and it consists of (4.2), (4.7), and 62 1 In 11 a2= 1--6t+e(~tal[-~a
2~ 0 1J+a 2
or d-~a=eQ0(a ) dt
If t > 0, an example for the first case is given by x2 5¢1= x2 5¢2=-x1-e t
Hence, the Struble solution consists of (4.2), (4.7), (4.10), and e2 a2=-~t+
Cl(a) } t
d a _ _ e _ _a 2 dt 2t
da _ _ e a l dt 2t
and the improved approximation is t l = a 0 { t } - e / 2 c o s t + a 02 {t} - e / 2 s .m t .
(4.12)
We finally note that the results of sections 2 and 3 can be extended to higher orders and the short periodic effects are used as a transformation set to yield corrections to the averaged equations. The justification of the perturbation method is given in [2] and therefore is not repeated here.
Each of the three approximations reduces to the same periodic solution when t -* ~ or when 0 a 2 = a 2 = 2. Table 1 indicates how closely in numerical agreement these various solutions are. The numerical integration is over the time interval [0, 1] with e = . 2. The quantities a x I and Ax 2 indicate the differences between numerical integration and the indicated approximation.
The author expresses his appreciation to prof. E. Cumberbatch of Purdue University for his help in preparing this paper. APPENDIX 1 - IDENTITIES
5. CONCLUSIONS
a In the derivations of section 2 and 3, the periodic nature of (1.1) enabled the averaged equations to be reduced to a simplified form. The spirit of the method, however, can be used independently of averaging on systems known to be a priori bounded. Consider equa-
~- aa }
aa
3F~
(A1) (A2)
C ( a ) : M t ()~p-~pF1)
Table 1. Comparison with numerical integration Xl(0)
Xl(1)
Numerical integration
.99747
.51735
Cole Kervorkian
.99747
Strubh D1 = D2 = 0
ax 1
x2(0)
x2(1)
0
-.15541
-.88529
0
.51503
.00232
-.15541
-.88118
-.00411
.99747
.51675
.0006
-.15541
-.88548
-.00019
.99747
.51565
.0017
-.15541
-.88246
-.0028-3
Journal of Computational and Applied Mathematics, volume 4, no 1, 1978.
ax 2
44
C(a) -- - aa ~ - ( ~a a
a
a___a_aa2~p ) -
771} + a---~ aa Fp1
aa
a~v a~pa~/3
a.__ { a_~.a} = _ aa
a2~v
aap
aa
aaaap
a~
a~v
a~p
a~fl
aa
aa
(A4)
(A5)
a2~0 a~-----v a~pa~ aa = a~v
a 2F0 v a~pa~/3
aa
+
a~/3 ~1 aa
~P
REFERENCES
1. POINCARI~, H. : "Sur les courbes par les 6quations diff& rentielles", J. de Math, (3) 7 (1886).
a 2F0 v
at ( a ~ p aaaa
a~
(A3)
aa (A6)
2. BOGOLIUBOV, MITROPOLSKY : "Asymptotic methods in the theory of non-linear oscillations", Gordon and Breach, New York, 1961. 3. COLE, KERVORKIAN : "Uniformly valid approximations for certain non-linear differential equations". Proceedings of the 1961 International Symposium on Non-Linear Differential Equations in Non-linear Mechanics, LaSalle and Lefschetz editors, Academic Press, New York, 1963, pp. 113-120. 4. STRUBLE, R. A. : "Non-linear differential equations", Mc Graw Hill, Inc. 1962. 5. MORRISON, J. A. : "Comparison of the modified method of averaging and the two variable expansion procedure", SIAM Review, Vol. 8, No. 1, 1966. 6. BURHSTEIN, SOLOV'EV : "Hamiltonian of averaged
motion", Soviet Physics Dokl, 6 (1962), pp. 731-733. 7. MELKA, R. F. : "Perturbation methods in ordinary differential equations", Ph.D. Thesis, Purdue Univ., August 1969.
Journal o f Computational and Applied Mathematics, volume 4, no 1, 1978.
45